We took DS for his first visit when he was only 7 weeks old, and the entire resort was out of 1st Visit buttons, which I have since been told happens quite often.
If City Hall is out try over at DCA. And, if that fails, it doesn't hurt to ask at each gift shop you visit or just to strike up conversations with cast members along the way as they sometimes have a few stashed away.
I was really sad when they were out and none of the normal places had them. However, a wonderful CM (named Mary I think) in Fantasyland came over to talk to us on our last day of the trip and inquired why DS didn't have a button as it was quite obviously his first trip. She magically produced a button for DS and 2 Honorary Citizen buttons for DH and me. She totally brightened our day with both the conversation and the unexpected gift!